{"id":"https://openalex.org/W7212061940","doi":"https://doi.org/10.48550/arxiv.2609.07657","title":"Classification of Solutions to a Critical Fourth-Order Equation on Complete Non-compact Manifolds","display_name":"Classification of Solutions to a Critical Fourth-Order Equation on Complete Non-compact Manifolds","publication_year":2026,"publication_date":"2026-09-07","ids":{"openalex":"https://openalex.org/W7212061940","doi":"https://doi.org/10.48550/arxiv.2609.07657"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.07657","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.07657","pdf_url":"https://arxiv.org/pdf/2609.07657","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.07657","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5159798436","display_name":"Huabin Li","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Li, Huabin","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5151072998","display_name":"Tian Wu","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Wu, Tian","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5001267178","display_name":"Xiao Zhou","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Zhou, Xiao","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.5439000129699707,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.5439000129699707,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.4514000117778778,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T13080","display_name":"Advanced Differential Geometry Research","score":0.00139999995008111,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/fourth-order-elliptic-equations","display_name":"fourth-order elliptic equations","score":0.9049999713897705},{"id":"https://openalex.org/keywords/noncompact-riemannian-manifolds","display_name":"non-compact Riemannian manifolds","score":0.8989999890327454},{"id":"https://openalex.org/keywords/nonnegative-ricci-curvature","display_name":"nonnegative Ricci curvature","score":0.9660000205039978},{"id":"https://openalex.org/keywords/rigidity","display_name":"rigidity","score":0.5759999752044678},{"id":"https://openalex.org/keywords/classification-of-solutions","display_name":"classification of solutions","score":0.7639999985694885},{"id":"https://openalex.org/keywords/continuity-method","display_name":"continuity method","score":0.8849999904632568},{"id":"https://openalex.org/keywords/invariant-tensors","display_name":"invariant tensors","score":0.9409999847412109},{"id":"https://openalex.org/keywords/differential-inequalities","display_name":"differential inequalities","score":0.6660000085830688},{"id":"https://openalex.org/keywords/finite-energy-solutions","display_name":"finite energy solutions","score":0.6299999952316284}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8608999848365784},{"id":"https://openalex.org/C2777984123","wikidata":"https://www.wikidata.org/wiki/Q9248237","display_name":"Pointwise","level":2,"score":0.6517999768257141},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5920000076293945},{"id":"https://openalex.org/C2779593128","wikidata":"https://www.wikidata.org/wiki/Q632814","display_name":"Riemannian manifold","level":2,"score":0.5831999778747559},{"id":"https://openalex.org/C145784156","wikidata":"https://www.wikidata.org/wiki/Q859808","display_name":"Biharmonic equation","level":3,"score":0.546500027179718},{"id":"https://openalex.org/C12089564","wikidata":"https://www.wikidata.org/wiki/Q1195879","display_name":"Ricci curvature","level":3,"score":0.5378999710083008},{"id":"https://openalex.org/C121705324","wikidata":"https://www.wikidata.org/wiki/Q1510587","display_name":"Pseudo-Riemannian manifold","level":4,"score":0.5353000164031982},{"id":"https://openalex.org/C20483540","wikidata":"https://www.wikidata.org/wiki/Q4200516","display_name":"Invariant manifold","level":2,"score":0.524399995803833},{"id":"https://openalex.org/C186450821","wikidata":"https://www.wikidata.org/wiki/Q17295","display_name":"Euclidean space","level":2,"score":0.5185999870300293},{"id":"https://openalex.org/C192939610","wikidata":"https://www.wikidata.org/wiki/Q188444","display_name":"Differential geometry","level":2,"score":0.51419997215271},{"id":"https://openalex.org/C74349738","wikidata":"https://www.wikidata.org/wiki/Q1517914","display_name":"Closed manifold","level":3,"score":0.4771000146865845},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.45329999923706055},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4490000009536743},{"id":"https://openalex.org/C160343418","wikidata":"https://www.wikidata.org/wiki/Q185256","display_name":"Rigidity (electromagnetism)","level":2,"score":0.4174000024795532},{"id":"https://openalex.org/C169391604","wikidata":"https://www.wikidata.org/wiki/Q7604402","display_name":"Statistical manifold","level":5,"score":0.4124999940395355},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.39719998836517334},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.3903000056743622},{"id":"https://openalex.org/C181104567","wikidata":"https://www.wikidata.org/wiki/Q761383","display_name":"Riemannian geometry","level":2,"score":0.35740000009536743},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.33399999141693115},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.3181999921798706},{"id":"https://openalex.org/C165818556","wikidata":"https://www.wikidata.org/wiki/Q213488","display_name":"Geodesic","level":2,"score":0.299699991941452},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.29100000858306885},{"id":"https://openalex.org/C2780390204","wikidata":"https://www.wikidata.org/wiki/Q1036595","display_name":"Blowing up","level":2,"score":0.275299996137619},{"id":"https://openalex.org/C91188154","wikidata":"https://www.wikidata.org/wiki/Q186247","display_name":"Vector field","level":2,"score":0.25589999556541443},{"id":"https://openalex.org/C115071613","wikidata":"https://www.wikidata.org/wiki/Q1047080","display_name":"Differential form","level":2,"score":0.25200000405311584}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.07657","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.07657","pdf_url":"https://arxiv.org/pdf/2609.07657","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.07657","doi":"https://doi.org/10.48550/arxiv.2609.07657","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.07657","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.07657","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.07657","pdf_url":"https://arxiv.org/pdf/2609.07657","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7212061940.pdf","grobid_xml":"https://content.openalex.org/works/W7212061940.grobid-xml"},"referenced_works_count":4,"referenced_works":["https://openalex.org/W4221161849","https://openalex.org/W4407570510","https://openalex.org/W4415239842","https://openalex.org/W4415239858"],"related_works":["https://openalex.org/W4415239842","https://openalex.org/W2036238812","https://openalex.org/W1991178943","https://openalex.org/W2051392474","https://openalex.org/W4407570510","https://openalex.org/W1992832037","https://openalex.org/W2028971690","https://openalex.org/W1535243070","https://openalex.org/W2964273223","https://openalex.org/W2107614808"],"abstract_inverted_index":{"We":[0,27],"study":[1],"the":[2,39,66,71,76],"critical":[3],"biharmonic":[4],"equation":[5],"\\[":[6,81],"\u0394^2":[7],"u=u^{\\frac{n+4}{n-4}},":[8],"\\]":[9,92],"on":[10],"a":[11,47,52,60],"complete,":[12],"connected,":[13],"and":[14,38,75],"non-compact":[15],"Riemannian":[16],"manifold":[17,67],"\\((M^n,g)\\)":[18],"of":[19],"dimension":[20],"\\(n\\geq":[21],"5\\)":[22],"with":[23],"non-negative":[24],"Ricci":[25],"curvature.":[26],"establish":[28],"an":[29],"optimal":[30],"pointwise":[31],"second-order":[32],"derivative":[33],"estimate":[34],"by":[35,80],"Bernstein's":[36],"method":[37],"continuity":[40],"method.":[41],"Using":[42],"invariant":[43],"tensors,":[44],"we":[45],"derive":[46],"differential":[48],"inequality":[49],"that":[50],"yields":[51],"rigidity":[53],"result.":[54],"More":[55],"precisely,":[56],"if":[57],"there":[58],"exists":[59],"positive":[61],"finite":[62],"energy":[63],"solution,":[64],"then":[65],"is":[68,78],"isometric":[69],"to":[70],"Euclidean":[72],"space":[73],"\\(\\mathbb{R}^n\\),":[74],"solution":[77],"given":[79],"u(x)=":[82],"\\frac{":[83],"\\left[\u03bb^2":[84],"n(n-4)(n^2-4)\\right]^{\\frac{n-4}{8}}":[85],"}":[86],"{\\left(1+\u03bb|x-x_0|^2\\right)^{\\frac{n-4}{2}}},":[87],"\\qquad":[88],"\u03bb>0,\\quad":[89],"x_0\\in\\mathbb{R}^n":[90],".":[91]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-10T00:00:00"}